s11 - Solution to Set 11 1. (a) 3 dr = t1/2 (i + j ) dt 2...

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Unformatted text preview: Solution to Set 11 1. (a) 3 dr = t1/2 (i + j ) dt 2 dr 3 F· = [x(t) + y (t)] t1/2 = 3t2 dt 2 C F · dr = 2 1 3t2 dt = 7 (b) dr = 2(− sin ti + cos tj ) dt dr F· = −2x(t) sin t + 2y (t) cos t = 0 dt C F · dr = 0 (c) dr 3 = t1/2 (i + j ) dt 2 y (t) − x(t) 3 1/2 dr =2 t F· =0 dt x (t) + y 2 (t) 2 C F · dr = 0 (d) dr = 2(− sin ti + cos tj ) dt dr 2y (t) sin t + 2x(t) cos t F· =− = −1 dt x2 (t) + y 2 (t) C F · dr = −2π Note that ∇ × F = 0 everywhere except at (0, 0) (that makes the line integral of the closed loop non-zero). 2. (a) Define the function f by performing a line integral along the line segments (0, 0) → (x, 0) and (x, 0) → (x, y ) x f (x, y ) = ∇f = (ex + y )i + (x + 2y )j = the field ∴ the field is conservative (0,1)→(2,3) 0 (et + 0)dt + y 0 (x + 2t)dt = ex + xy + y 2 ⇒ C F · dr is path independent. F · dr = e2 + 2 · 3 + 32 − (e0 + 0 + 1) = e2 + 13 (b) Define the function f by performing a line integral along the line segments (0, 0) → (x, 0) and (x, 0) → (x, y ) x y f (x, y ) = 0 (0 + 1)dt + 0 2x2 tdt = x + x2 y 2 ∇f = (1 + 2xy 2 )i + 2x2 y j = the field ∴ the field is conservative (−1,2)→(2,3) ⇒ C F · dr is path independent. F · dr = 2 + 4 · 9 − (−1 + 4) = 35 1 3. (a) Let R be the region enclosed by the closed curve C . C F · dr = R (∂x 3x − ∂y y )dA = R 2dA = 8π (b) Let R be the region enclosed by the closed curve C . C F · dr = R (∂x x3 − ∂y y 4 )dA = 4 x3 2 −2 − 4 y4 2 −2 = 64 2 ...
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This note was uploaded on 02/27/2010 for the course MATH MATH101 taught by Professor Chan during the Fall '09 term at HKUST.

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s11 - Solution to Set 11 1. (a) 3 dr = t1/2 (i + j ) dt 2...

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